A Logical Approach to Efficient Max-SAT solving

Weighted Max-SAT is the optimization version of SAT and many important problems can be naturally encoded as such. Solving weighted Max-SAT is an important problem from both a theoretical and a practical point of view. In recent years, there has been considerable interest in finding efficient solving techniques. Most of this work focus on the computation of good quality lower bounds to be used within a branch and bound DPLL-like algorithm. Most often, these lower bounds are described in a procedural way. Because of that, it is difficult to realize the {\em logic} that is behind. In this paper we introduce an original framework for Max-SAT that stresses the parallelism with classical SAT. Then, we extend the two basic SAT solving techniques: {\em search} and {\em inference}. We show that many algorithmic {\em tricks} used in state-of-the-art Max-SAT solvers are easily expressable in {\em logic} terms with our framework in a unified manner. Besides, we introduce an original search algorithm that performs a restricted amount of {\em weighted resolution} at each visited node. We empirically compare our algorithm with a variety of solving alternatives on several benchmarks. Our experiments, which constitute to the best of our knowledge the most comprehensive Max-sat evaluation ever reported, show that our algorithm is generally orders of magnitude faster than any competitor

Cost Function Networks to Solve Large Computational Protein Design Problems

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Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization

Cost Function Networks to Solve Large Computational Protein Design Problems

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Virtual camera selection using a semiring constraint satisfaction approach

Players and viewers of three-dimensional computer generated games and worlds view renderings from the viewpoint of a virtual camera. As such, determining a good view of the scene is important to present a good game or three-dimensional world. Previous research has developed technologies to nd good positions for the virtual camera, but little work has been done to automatically select between multiple virtual cameras, similar to a human director at a sporting event. This thesis describes a software tool to select among camera feeds from multiple virtual cameras in a virtual environment using semiring-based constraint satisfaction techniques (SCSP), a soft constraint approach. The system encodes a designer's preferences, and selects the best camera feed even in over-constrained or under-constrained environments. The system functions in real time for dynamic scenes using only current information (i.e. no prediction). To reduce the camera selection time the SCSP evaluation can be cached and converted to native code. This SCSP approach is implemented in two virtual environments: a virtual hockey game using a spectator viewpoint, and a virtual 3D maze game using a third person perspective. Comparisons against hard constraints are made using constraint satisfaction problems

Tractable projection-safe soft global constraints in weighted constraint satisfaction.

Wu, Yi.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 74-80).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.3Chapter 1.3 --- Motivation and Goal --- p.4Chapter 1.4 --- Outline of the Thesis --- p.5Chapter 2 --- Background --- p.7Chapter 2.1 --- Constraint Satisfaction Problems --- p.7Chapter 2.1.1 --- Backtracking Tree search --- p.8Chapter 2.1.2 --- Local consistencies in CSP --- p.11Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.18Chapter 2.2.1 --- Branch and Bound Search --- p.20Chapter 2.2.2 --- Local Consistencies in WCSP --- p.21Chapter 2.3 --- Global Constraints --- p.31Chapter 3 --- Tractable Projection-Safety --- p.36Chapter 3.1 --- Tractable Projection-Safety: Definition and Analysis --- p.37Chapter 3.2 --- Polynomially Decomposable Soft Constraints --- p.42Chapter 4 --- Examples of Polynomially Decomposable Soft Global Constraints --- p.48Chapter 4.1 --- Soft Among Constraint --- p.49Chapter 4.2 --- Soft Regular Constraint --- p.51Chapter 4.3 --- Soft Grammar Constraint --- p.54Chapter 4.4 --- Max_Weight/Min Weight Constraint --- p.57Chapter 5 --- Experiments --- p.61Chapter 5.1 --- The car Sequencing Problem --- p.61Chapter 5.2 --- The nonogram problem --- p.62Chapter 5.3 --- Well-Formed Parenthesis --- p.64Chapter 5.4 --- Minimum Energy Broadcasting Problem --- p.64Chapter 6 --- Related Work --- p.67Chapter 6.1 --- WCSP Consistencies --- p.67Chapter 6.2 --- Global Constraints . --- p.68Chapter 7 --- Conclusion --- p.71Chapter 7.1 --- Contributions --- p.71Chapter 7.2 --- Future Work --- p.72Bibliography --- p.7

Multi-objective optimization in graphical models

Many real-life optimization problems are combinatorial, i.e. they concern a choice of the best solution from a finite but exponentially large set of alternatives. Besides, the solution quality of many of these problems can often be evaluated from several points of view (a.k.a. criteria). In that case, each criterion may be described by a different objective function. Some important and well-known multicriteria scenarios are: · In investment optimization one wants to minimize risk and maximize benefits. · In travel scheduling one wants to minimize time and cost. · In circuit design one wants to minimize circuit area, energy consumption and maximize speed. · In knapsack problems one wants to minimize load weight and/or volume and maximize its economical value. The previous examples illustrate that, in many cases, these multiple criteria are incommensurate (i.e., it is difficult or impossible to combine them into a single criterion) and conflicting (i.e., solutions that are good with respect one criterion are likely to be bad with respect to another). Taking into account simultaneously the different criteria is not trivial and several notions of optimality have been proposed. Independently of the chosen notion of optimality, computing optimal solutions represents an important current research challenge. Graphical models are a knowledge representation tool widely used in the Artificial Intelligence field. They seem to be specially suitable for combinatorial problems. Roughly, graphical models are graphs in which nodes represent variables and the (lack of) arcs represent conditional independence assumptions. In addition to the graph structure, it is necessary to specify its micro-structure which tells how particular combinations of instantiations of interdependent variables interact. The graphical model framework provides a unifying way to model a broad spectrum of systems and a collection of general algorithms to efficiently solve them. In this Thesis we integrate multi-objective optimization problems into the graphical model paradigm and study how algorithmic techniques developed in the graphical model context can be extended to multi-objective optimization problems. As we show, multiobjective optimization problems can be formalized as a particular case of graphical models using the semiring-based framework. It is, to the best of our knowledge, the first time that graphical models in general, and semiring-based problems in particular are used to model an optimization problem in which the objective function is partially ordered. Moreover, we show that most of the solving techniques for mono-objective optimization problems can be naturally extended to the multi-objective context. The result of our work is the mathematical formalization of multi-objective optimization problems and the development of a set of multiobjective solving algorithms that have been proved to be efficient in a number of benchmarks.Muchos problemas reales de optimización son combinatorios, es decir, requieren de la elección de la mejor solución (o solución óptima) dentro de un conjunto finito pero exponencialmente grande de alternativas. Además, la mejor solución de muchos de estos problemas es, a menudo, evaluada desde varios puntos de vista (también llamados criterios). Es este caso, cada criterio puede ser descrito por una función objetivo. Algunos escenarios multi-objetivo importantes y bien conocidos son los siguientes: · En optimización de inversiones se pretende minimizar los riesgos y maximizar los beneficios. · En la programación de viajes se quiere reducir el tiempo de viaje y los costes. · En el diseño de circuitos se quiere reducir al mínimo la zona ocupada del circuito, el consumo de energía y maximizar la velocidad. · En los problemas de la mochila se quiere minimizar el peso de la carga y/o el volumen y maximizar su valor económico. Los ejemplos anteriores muestran que, en muchos casos, estos criterios son inconmensurables (es decir, es difícil o imposible combinar todos ellos en un único criterio) y están en conflicto (es decir, soluciones que son buenas con respecto a un criterio es probable que sean malas con respecto a otra). Tener en cuenta de forma simultánea todos estos criterios no es trivial y para ello se han propuesto diferentes nociones de optimalidad. Independientemente del concepto de optimalidad elegido, el cómputo de soluciones óptimas representa un importante desafío para la investigación actual. Los modelos gráficos son una herramienta para la represetanción del conocimiento ampliamente utilizados en el campo de la Inteligencia Artificial que parecen especialmente indicados en problemas combinatorios. A grandes rasgos, los modelos gráficos son grafos en los que los nodos representan variables y la (falta de) arcos representa la interdepencia entre variables. Además de la estructura gráfica, es necesario especificar su (micro-estructura) que indica cómo interactúan instanciaciones concretas de variables interdependientes. Los modelos gráficos proporcionan un marco capaz de unificar el modelado de un espectro amplio de sistemas y un conjunto de algoritmos generales capaces de resolverlos eficientemente. En esta tesis integramos problemas de optimización multi-objetivo en el contexto de los modelos gráficos y estudiamos cómo diversas técnicas algorítmicas desarrolladas dentro del marco de los modelos gráficos se pueden extender a problemas de optimización multi-objetivo. Como mostramos, este tipo de problemas se pueden formalizar como un caso particular de modelo gráfico usando el paradigma basado en semi-anillos (SCSP). Desde nuestro conocimiento, ésta es la primera vez que los modelos gráficos en general, y el paradigma basado en semi-anillos en particular, se usan para modelar un problema de optimización cuya función objetivo está parcialmente ordenada. Además, mostramos que la mayoría de técnicas para resolver problemas monoobjetivo se pueden extender de forma natural al contexto multi-objetivo. El resultado de nuestro trabajo es la formalización matemática de problemas de optimización multi-objetivo y el desarrollo de un conjunto de algoritmos capaces de resolver este tipo de problemas. Además, demostramos que estos algoritmos son eficientes en un conjunto determinado de benchmarks

Soft global constraints in constraint optimization and weighted constraint satisfaction.

Leung, Ka Lun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 118-126).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction and Global Constraints --- p.3Chapter 1.2 --- Soft Constraints --- p.4Chapter 1.3 --- Motivation and Goal --- p.5Chapter 1.4 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.10Chapter 2.1.2 --- Local Consistency in CSP --- p.11Chapter 2.1.3 --- Constraint Optimization Problem --- p.16Chapter 2.2 --- Weighted Constraint Satisfaction --- p.21Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistency in WCSP --- p.26Chapter 2.3 --- Global Constraints --- p.35Chapter 2.4 --- Flow Theory --- p.37Chapter 3 --- Related Work --- p.39Chapter 3.1 --- Handling Soft Constraints in COPs --- p.39Chapter 3.2 --- Global Constraints --- p.40Chapter 3.2.1 --- Hard Global Constraints --- p.40Chapter 3.2.2 --- Soft Global Constraints --- p.41Chapter 3.3 --- Local Consistency in Weighted CSP --- p.42Chapter 4 --- “Soft as Hard´ح Approach --- p.44Chapter 4.1 --- The General “Soft as Hard´ح Approach --- p.44Chapter 4.2 --- Cost-based GAC --- p.49Chapter 4.3 --- Empirical Results --- p.53Chapter 5 --- Weighted CSP Approach --- p.55Chapter 5.1 --- Strong 0-Inverse Consistency --- p.55Chapter 5.1.1 --- 0-Inverse Consistency and Strong 0-Inverse Consistency --- p.56Chapter 5.1.2 --- Comparison with Other Consistencies --- p.62Chapter 5.2 --- Generalized Arc Consistency Star --- p.65Chapter 5.3 --- Full Directional Generalized Arc Consistency Star --- p.72Chapter 5.4 --- Generalizing EDAC* --- p.78Chapter 5.5 --- Implementation Issues --- p.87Chapter 6 --- Towards A Library of Efficient Soft Global Constraints --- p.90Chapter 6.1 --- The allDifferent Constraint --- p.91Chapter 6.1.1 --- All Interval Series --- p.93Chapter 6.1.2 --- Latin Square --- p.95Chapter 6.2 --- The GCC Constraint --- p.97Chapter 6.2.1 --- Latin Square --- p.100Chapter 6.2.2 --- Round Robin Tournament --- p.100Chapter 6.3 --- The Same Constraint --- p.102Chapter 6.3.1 --- Fair Scheduling --- p.104Chapter 6.3.2 --- People-Mission Scheduling --- p.105Chapter 6.4 --- The Regular Constraint --- p.106Chapter 6.4.1 --- Nurse Rostering Problem --- p.110Chapter 6.4.2 --- Modelling Stretch() Constraint --- p.111Chapter 6.5 --- Discussion --- p.113Chapter 7 --- Conclusion and Remarks --- p.115Chapter 7.1 --- Contributions --- p.115Chapter 7.2 --- Future Work --- p.117Bibliography --- p.11

Strong consistencies for weighted constraint satisfaction problems

Cette thèse se focalise sur l'étude de cohérences locales fortes afin de résoudre des problèmes d'optimisation sur des réseaux de fonctions de coûts (ou réseaux de contraintes pondérées). Ces méthodes fournissent le minorant nécessaire pour des approches de type "Séparation-Evaluation". Nous étudions dans un premier temps la cohérence d'Arc virtuelle (VAC), une des plus fortes cohérences d'arcs du domaine, qui est établie via l'établissement de la cohérence d'arc dure dans une séquence de réseaux de contraintes classiques. L'algorithme itératif pour établir VAC est amélioré via l'introduction d'une incrémentalité accrue, exploitant la cohérence d'arc dynamique. La nouvelle méthode est aussi capable de maintenir VAC efficacement pendant la recherche lorsque les réseaux de contraintes pondérées sont dynamiquement modifiés par les opérations de branchement. Dans une seconde partie, nous nous intéressons à des cohérences de domaines plus fortes, inspirées de cohérences similaires dans les réseaux de contraintes classiques (cohérence de chemin inverse, réduite ou Max-réduite). Pour chaque cohérence dure, plusieurs cohérences souples ont été proposées pour les réseaux de contraintes pondérées. Les nouvelles cohérences fournissent un minorant plus fort que celui des cohérences d'arc souples en traitant les triplets de variables connectées deux à deux par des fonctions de coûts binaires. Dans cette thèse, nous étudions les propriétés des nouvelles cohérences, les implémentons et les testons sur une variété de problèmes.This thesis focuses on strong local consistencies for solving optimization problems in cost function networks (or weighted constraint networks). These methods provide the lower bound necessary for Branch-and-Bound search. We first study the Virtual arc consistency, one of the strongest soft arc consistencies, which is enforced by iteratively establishing hard arc consistency in a sequence of classical Constraint Networks. The algorithm enforcing VAC is improved by integrating the dynamic arc consistency to exploit its incremental behavior. The dynamic arc consistency also allows to improve VAC when maintained VAC during search by efficiently exploiting the changes caused by branching operations. Operations. Secondly, we are interested in stronger domain-based soft consistencies, inspired from similar consistencies in hard constraint networks (path inverse consistency, restricted or Max-restricted path consistencies). From each of these hard consistencies, many soft variants have been proposed for weighted constraint networks. The new consistencies provide lower bounds stronger than soft arc consistencies by processing triplets of variables connected two-by-two by binary cost functions. We have studied the properties of these new consistencies, implemented and tested them on a variety of problems

High-order consistency in valued constraint satisfaction

k-consistency operations in constraint satisfaction problems (CSPs) render constraints more explicit by solving size-k subproblems and projecting the information thus obtained down to low-order constraints. We generalise this notion of k-consistency to valued constraint satisfaction problems (VCSPs) and show that it can be established in polynomial time when penalties lie in a discrete valuation structure. A generic definition of consistency is given which can be tailored to particular applications. As an example, a version of high-order consistency (face consistency) is presented which can be established in low-order polynomial time given certain restrictions on the valuation structure and the form of the constraint graph